INDET'ERMINATE PROBLEMS. 199 



be nearly equal, the approximation will be more accurate. 

 Let m :zi a; then the denominator ni^ — ab will be fmall, and 

 therefore the fradtions large ; whence, by fubftitution, 



^a-^-b : ^b-\-a, nearly 



: : in*? : TOO : : T : *7-J- 



Thus ^49 : v/50 '■ •• 197 '• 199 ■• ■ 7 ■• 7+I7-. whence /50 =:: 



7,07107, true to the laft place. 



Cor. 2. Let m =. — — ; thenm^ — abzz [— j — ab =: \~) > 

 which, when a and b are nearly equal, will be fmall, and by 



i- — j +ab+a{ti+l>) ( \ +ab+b{a+b) 



fubftitution, ^/a :^b:: , . : ■ — T^Zjip:^ , nearly 5 



hence, by proper redudlions, ^a : Vb : : ^a?-\-\oab-\-b'^ : 5^--{- 

 lo^3+<^^ This formula is more intricate than the former, but 

 ftill more accurate. Thus, ^9 : /lo : : 405+900-I-100 : 500+ 

 900+81 ■=. 1405 : 1481, and •/lo ■=. 3,16209, true to the laf^ 

 place. 



PROBLEM VI. 



Let it be required to find a mtnd^er, fuch that, if given multiples 

 of it be increafed by given numbers, the froduS. of the fums pall he 

 a fquare. 



Let [ex-^-f) {gx-\-h') zz y'^ ; by affumption ex-{f — 7ny, and 



gx-\-h — -^. Tranfpofing the firft equation, and dividing, 



W V f 



X zz — =-^ — . Reducing the fecond, nigx-\-mh ■=. y, and tranfpo- 

 fing 



